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Prof. A. Meher Prasad
   Department of Civil Engineering
Indian Institute of Technology Madras
     email: prasadam@iitm.a...
Typical Accelerograms




From Dynamics of Structures by
A K Chopra, Prentice Hall
                                 Time, sec
Response Spectrum


•   If the ground moves as per the given accelerogram, what is the
    maximum response of a single de...
Response Spectrum (contd…)



•   Using a computer, one can calculate the response of SDOF system
    with time (time hist...
Response Spectrum (contd…)

Maximum response = 7.47 in.

                                       T=2 sec,
   U(t)          ...
Response Spectrum (contd…)



•   Repeat this exercise for different values of natural period.

•   For design, we usually...
Response Spectrum (contd…)

                                 Displacement Response
                                 Spectr...
RESPONSE SPECTRUM – IS 1893:2002
Response Spectrum (contd…)


      • Different terms used in the code:

         - Design Acceleration Spectrum (clause 3....
Smooth Response Spectrum



  • Real spectrum has somewhat irregular shape with local peaks
    and valleys

  • For desig...
Smooth Response Spectrum (contd…)




      Period (sec)              Period (sec)              Period (sec)
  Acceleratio...
Floor Response Spectrum



•   Equipment located on a floor needs to be designed for the motion
    experienced by the flo...
Response Spectrum versus Design Spectrum



•   Consider the Acceleration Response Spectrum
•   Notice the region of red c...
Response Spectrum versus Design Spectrum (contd…)



 •   Natural period of a civil engineering structure cannot be calcul...
Design Spectrum



•   Since some damage is expected and accepted in the structure
    during strong shaking, design spect...
Design Spectrum (contd…)

 •   Design Spectrum must be accompanied by:

     – Load factors or permissible stresses that m...
Design Spectrum (contd…)


•   1984 code provided slightly different design spectrum for two
    methods
     – Seismic Co...
IS:1893-1984


•   Design base shear for a building by Seismic Coefficient Method was
    calculated as
        Vb= oIKC...
IS:1893-1984 (contd…)



•   In the Response Spectrum Method, the design spectrum was given
    by FoIK(Sa/g)
           ...
Major Changes in Design Spectrum



  •   Zone Factor (Z) is specified in place of o and Fo
  •   Importance Factor (I) i...
Soil Effect


 •   Recorded earthquake motions show that response spectrum shape
     differs for different type of soil p...
Shape of Design Spectrum



 •   The three curves in Fig. 2 have been drawn based on general
     trends of average respon...
IS1893:2002

 Local soil profile reflected through a different design spectrum for Rock , Soil
 Normalized for Peak Grou...
BACKGROUND

Discussed in SDOF System
Spectral Quantities…


 This may also be viewed as the equivalent lateral static force which
 produces the same effects as...
Spectral Quantities…


Another useful measure of the maximum deformation, U, is the pseudo
velocity of the system, defined...
Deformation spectra




         1.Obtained from results already presented
         2.Presentation of results in alternate...
General form of spectrum

                                                      ..
It approaches U = y0 at extreme left; a...
Generation of results


• General form of spectrum is as shown in next slide

 (a) It approaches V= y0 at the extreme left...
General form of spectrum
Acceleration spectra for elastic
system - El Centro Earthquake
SDF systems with 10%
                                             damping subjected to El
                                ...
Spectral Regions


The characteristics of the ground motion which control the deformation of
SOF systems are different for...
Spectral Regions…


Minor differences in these characteristics may have a significant effect on
the magnitude of the defor...
Spectral Regions…



    The boundaries of the various frequency regions are different
   for different excitations and, f...
Logarithmic plot of Deformation Spectra

It is convenient to display the spectra or a log-log paper, with the
abscissa rep...
Logarithmic plot of Deformation Spectra

                                Velocity
                                sensitiv...
Deformation Spectra for Half-cycle Acceleration pulse:

 This class of excitation is associated with a finite terminal vel...
t1

Letting P(t )   m &&(t ) and
                    y              X max  U and noting that
                          ...
Spectra for maximum and minimum accelerations of the mass
    (undamped elastic systems subjected to a Half cycle
        ...
Spectra for maximum and minimum acceleration of the mass
(undamped Elastic systems subjected to a versed-sine velocity
   ...
Deformation spectra for undamped elastic systems
     subjected to a versed-sine velocity pulse
‘B’ Level Earthquake (=10% ; μ=1.0)
Deformation spectrum for undamped Elastic systems
     subjected to a half-sine acceleration pulse
Example:


For a SDF undamped system with a natural frequency,f=2cps,evaluate
the maximum value of the deformation,U when ...
Therefore
        2 ..     2      1
 2p fU ᄏ f1 y 0  ᄡ 0.1ᄡ ᄡ 9.81
        p        p      2
      1 0.1 1
 U ᄏ 2 ᄡ ᄡ ᄡ 9...
A 0.8  1 2  g
Accordingly, C                  0.4
                 g       g
             Q0  0.4W
                 ...
If the duration of the pulse were t1 = 0.75 sec instead of 0.1 sec , the
results would be as follows

           ft1  2 ᄡ...
If the duration of the pulse were t1,as in the first case, but the natural
frequency of the system were 15 cps instead of ...
• Plot spectra for inputs considered in the illustrative example and compare

                          y0 For t1=0.75sec
...
Design Spectrum

                                                               xmax         A
May be determined from the ...
Deformation Spectra for Half-Cycle Velocity Pulses


 Refer to spectrum for   0
 Note the following
 •   At extreme righ...
be conservative. Improved estimate may be obtained by considering
relative durations of the individual pulses and superpos...
It should be clear that,

(c) The left-hand, inclined portion of the spectrum to displacement
    sensitive.

(e) The midd...
Design Rules




     Design spectrum for the absolute maximum deformation of
     systems subjected to a half cycle veloc...
Deformation spectra for undamped elastic systems
subjected to skewed versed-sine velocity pulses
Deformation Spectra for Half-cycle Displacement Pulse

See spectrum for undamped systems, =0, on the next page

Note that...
Design Rules




     Design spectrum for maximum deformation of systems
     subjected to a half cycle displacement pulse
However the spectrum is no longer bounded on the left by the line
U= yo, but exhibits a hump with peak value of U0 = 1.6 y...
Deformation spectra for damped elastic systems subjected
to a half cycle displacement pulse
Deformation spectra for full cycle Displacement pulse
As would be expected ,the maximum value of U in this case is
approximately 3.2 yo .Furthermore, the left hand portion of t...
Effect of Discontinuous Acceleration Pulses

The high frequency limit of the deformation spectrum is sensitive to
whether ...
Deformation spectra for damped elastic systems subjected
to a full cycle displacement pulse
Application to Complex Ground Motions

• Compound Pulses
• Earthquake Records
      Eureka record
      El-Centro record

...
Effect of damping:


• Effect is different in different frequency ranges
• Effect is negligible in the extremely low frequ...
Eureka, California earthquake of Dec 21,1954 S 11o E
component.
Eureka Quake
Elcentro ,California Earthquake of May 18,1940,N-S component
V
                                       = pseudo velocity
                                     Yc Maximum Ground Velocity...
Further discussion of Design Response Spectra

  The specification of the design spectrum by the procedure that has
  been...
1. Estimating the maximum spectral amplification factors, αD, αV, αA ;
  for the various parts of the spectrum.

   Again ...
Following are the values proposed in a recent unpublished paper
by Newmark & Hall for horizontal motions:

 Damping       ...
Ground Acceleration


•   Number of empirical relations available in literature to correlate
    shaking intensity with Pe...
Table


Average horizontal peak ground acceleration as a function of earthquake intensity

 Intensity (MM                 ...
Ground Acceleration


•   ZPA stands for Zero Period Acceleration.
     – Implies max acceleration experienced by a struct...
Example: Determine the response spectrum for a design earthquake
with &&  0.3g ye  0.3 m / sec and y0  0.25 m. Take  ...
Design Earthquakes
Describing the Earthquake

Ground Motion Time Histories

 Ground motion time histories are numerical d...
Statistically Derived Design Spectra

 The general procedure for generating statistically derived spectra is as follows:
...
Effect of various factors on spectral values
Soil Conditions

 For soft soils, ag remains the same or
 decreases relative...
Effect of various factors on spectral values
Near Fault Motions and Fault Rupture Directivity
For near-fault motions ag in...
Effect of various factors on spectral values
Near Fault Motions and Fault Rupture Directivity (Cont..)


 Somerville et al...
Effect of various factors on spectral values
Viscous Damping


Friction between and with structural and non-structural ele...
Effect of various factors on spectral values
Viscous Damping

  Viscous Damping Values for Design

  Many codes stipulat...
Effect of Various Factors on Spectral Values
Modifying the Viscous Damping of Spectra

  Newmark and Hall's Method

  For ...
Role of Ductility
Elasto Plastic Force   Elasto Plastic system and its
Deformation relation   corresponding linear system
Design Values of normalized yield Strength
Construction of Elastic Design Spectrum
Construction of Inelastic Design Spectrum
Response of Elastoplastic system to Elcentro Ground motion
Empirically Derived Design Spectra
Basic Concepts

  The complexity of the previous methods, and the limited number of
  ...
Empirically Derived Design Spectra
 Newmark and Hall's Method

 N. M. Newmark and W. J. Hall's procedure
  for developing...
Empirically Derived Design Spectra (Cont..)
Newmark and Hall's Method

 Structural Response Amplification Factors

       ...
Empirically Derived Design Spectra (Cont..)
Newmark and Hall's Method

Tripartite Plots:

Newmark and Hall's spectra are p...
Empirically Derived Design Spectra (Cont..)
Newmark and Hall's Method


 Then spectral acceleration and spectral displace...
Empirically Derived Design Spectra (Cont..)
Newmark and Hall's Method

  All three types of spectrum (Sa vs. T, Sv vs. T,...
Empirically Derived Design Spectra
Constructing Newmark and Hall Spectra


1. Construct ground motion 'backbone' curve usi...
Response Spectrum
Response Spectrum
Response Spectrum
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Response Spectrum

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Response Spectrum

  1. 1. Prof. A. Meher Prasad Department of Civil Engineering Indian Institute of Technology Madras email: prasadam@iitm.ac.in
  2. 2. Typical Accelerograms From Dynamics of Structures by A K Chopra, Prentice Hall Time, sec
  3. 3. Response Spectrum • If the ground moves as per the given accelerogram, what is the maximum response of a single degree of freedom (SDOF) system (of given natural period and damping)? – Response may mean any quantity of interest, e.g., deformation, acceleration a(t)/g T=2 sec, Damping  =2% Time, sec Ground motion time history
  4. 4. Response Spectrum (contd…) • Using a computer, one can calculate the response of SDOF system with time (time history of response) • Can pick maximum response of this SDOF system (of given T and damping) from this response time history – See next slide
  5. 5. Response Spectrum (contd…) Maximum response = 7.47 in. T=2 sec, U(t) Damping  =2% Time, sec Time History of Deformation (relative displacement of mass with respect to base) response A(t)/g Time, sec Ground motion time history
  6. 6. Response Spectrum (contd…) • Repeat this exercise for different values of natural period. • For design, we usually need only the maximum response. • Hence, for future use, plot maximum response versus natural period (for a given value of damping). • Such a plot of maximum response versus natural period for a given accelerogram is called response spectrum.
  7. 7. Response Spectrum (contd…) Displacement Response Spectrum for the above time A(t)/g history Time, sec T=0.5 sec U(t)  =2% T=1.0 sec U(t)  =2% Umax T=2.0 sec U(t)  =2% Time, sec T, sec Figure After Chopra, 2001
  8. 8. RESPONSE SPECTRUM – IS 1893:2002
  9. 9. Response Spectrum (contd…) • Different terms used in the code: - Design Acceleration Spectrum (clause 3.5) – Response Spectrum (clause 3.27) – Acceleration Response Spectrum (used in cl. 3.30) – Design Spectrum (title of cl. 6.4) – Structural Response Factor – Average response acceleration coefficient (see terminology of Sa/g on p. 11) – Title of Fig. 2: Response Spectra for ….
  10. 10. Smooth Response Spectrum • Real spectrum has somewhat irregular shape with local peaks and valleys • For design purpose, local peaks and valleys should be ignored – Since natural period cannot be calculated with that much accuracy. • Hence, smooth response spectrum used for design purposes • For developing design spectra, one also needs to consider other issues.
  11. 11. Smooth Response Spectrum (contd…) Period (sec) Period (sec) Period (sec) Acceleration Spectra Velocity Spectra Displacement Spectra Shown here are typical smooth spectra used in design for different values of damping (Fig. from Housner, 1970)
  12. 12. Floor Response Spectrum • Equipment located on a floor needs to be designed for the motion experienced by the floor. • Hence, the procedure for equipment will be: – Analyze the building for the ground motion. – Obtain response of the floor. – Express the floor response in terms of spectrum (termed as Floor Response Spectrum) – Design the equipment and its connections with the floor as per Floor Response Spectrum.
  13. 13. Response Spectrum versus Design Spectrum • Consider the Acceleration Response Spectrum • Notice the region of red circle marked: a slight change in natural period can lead to large variation in maximum acceleration Spectral Acceleration, g Undamped Natural Period T (sec)
  14. 14. Response Spectrum versus Design Spectrum (contd…) • Natural period of a civil engineering structure cannot be calculated precisely • Design specification should not very sensitive to a small change in natural period. • Hence, design spectrum is a smooth or average shape without local peaks and valleys you see in the response spectrum
  15. 15. Design Spectrum • Since some damage is expected and accepted in the structure during strong shaking, design spectrum is developed considering the overstrength, redundancy, and ductility in the structure. • The site may be prone to shaking from large but distant earthquakes as well as from medium but nearby earthquakes: design spectrum may account for these as well. – See Fig. next slide.
  16. 16. Design Spectrum (contd…) • Design Spectrum must be accompanied by: – Load factors or permissible stresses that must be used • Different choice of load factors will give different seismic safety to the structure – Damping to be used in design • Variation in the value of damping used will affect the design force. – Method of calculation of natural period • Depending on modeling assumptions, one can get different values of natural period. – Type of detailing for ductility • Design force can be lowered if structure has higher ductility.
  17. 17. Design Spectrum (contd…) • 1984 code provided slightly different design spectrum for two methods – Seismic Coefficient Method (static method), and – Response Spectrum Method (dynamic method) • It was confusing to use two different sets of terminology for two methods. • Present code provides same design spectrum irrespective of whether static or dynamic method is used.
  18. 18. IS:1893-1984 • Design base shear for a building by Seismic Coefficient Method was calculated as Vb= oIKCW C Natural Period (sec) • In a way, one could say that the design spectrum for the seismic coefficient method in the 1984 code was given by oIKC
  19. 19. IS:1893-1984 (contd…) • In the Response Spectrum Method, the design spectrum was given by FoIK(Sa/g) Sa/g = Average Acceleration Coefficient Natural Period (sec)
  20. 20. Major Changes in Design Spectrum • Zone Factor (Z) is specified in place of o and Fo • Importance Factor (I) is same • Soil Effect is considered by different shapes of response spectrum; Soil-Foundation Factor () has now been dropped. • Response Reduction Factor (R) used in denominator; earlier Performance Factor (K) was used in numerator. – For more ductile structures, K was lower. – Now, R will be higher for more ductile structures. • Structure Flexibility Factor (Sa/g); earlier C or Sa/g
  21. 21. Soil Effect • Recorded earthquake motions show that response spectrum shape differs for different type of soil profile at the site Period (sec) Fig. from Geotechnical Earthquake Engineering, by Kramer, 1996
  22. 22. Shape of Design Spectrum • The three curves in Fig. 2 have been drawn based on general trends of average response spectra shapes. • In recent years, the US codes (UBC, NEHRP and IBC) have provided more sophistication wherein the shape of design spectrum varies from area to area depending on the ground motion characteristics expected.
  23. 23. IS1893:2002  Local soil profile reflected through a different design spectrum for Rock , Soil  Normalized for Peak Ground Acceleration (PGA) of 1.0 Rocky or hard sites, 1 + 15 T 0.00 ≤ T ≤ 0.10 Sa / g = 2.50 0.10 ≤ T ≤ 0.40 1.00 / T 0.40 ≤ T ≤ 4.00 Medium soil sites 1 + 15 T 0.00 ≤ T ≤ 0.10 Damping 5% Sa / g = 2.50 0.10 ≤ T ≤ 0.55 1.36 / T 0.55 ≤ T ≤ 4.00 Soft soil sites 1 + 15 T 0.00 ≤ T ≤ 0.10 Sa / g = 2.50 0.10 ≤ T ≤ 0.67 1.67 / T 0.67 ≤ T ≤ 4.00 Damping 0 2 5 7 10 15 20 25 30 percent Factors 3.2 1.4 1.00 0.90 0.80 0.70 0.60 0.55 0.50 (new code)
  24. 24. BACKGROUND Discussed in SDOF System
  25. 25. Spectral Quantities… This may also be viewed as the equivalent lateral static force which produces the same effects as the maximum effects by the ground shaking. It is sometimes convenient to express Qmax in the form , Qmax  CW (B18) Where W = mg is the weight of the system. The quantity C is the so called lateral force coefficient, which represents the number of times the system must be capable of supporting its weight in the direction of motion. From Eqn.B17 and B18 it follows that, C=A/g (B19)
  26. 26. Spectral Quantities… Another useful measure of the maximum deformation, U, is the pseudo velocity of the system, defined as, V = p U (B20) The maximum strain energy stored in the spring can be expressed in terms of V as follows: Emax = (1/2) (k U) U = (1/2) m(pU)2 = (1/2)mV2 (B21) Under certain conditions, that we need not go into here, V is identical to ,or approximately equal to the maximum values of the relative velocity of the mass and the bays, U and the two quantities can be used interchangeably. However this is not true in general, and care should be exercised in replacing one for the other.
  27. 27. Deformation spectra 1.Obtained from results already presented 2.Presentation of results in alternate forms (a) In terms of U (b) In terms of V (c) In terms of A 3.Tripartite Logarithmic Plot
  28. 28. General form of spectrum .. It approaches U = y0 at extreme left; a value of A  y0 extreme right; It exhibits a hump on either side of the nearly horizontal central portion;and attains maximum values of U, V and .. which may be . A materially greater than the values of y0 , y 0 , and y0 It is assumed that the acceleration trace of the ground motion,and hence the associated velocity and displacement traces, are smooth continuous functions. The high-frequency limit of the response spectrum for discontinuous acceleration inputs may be significantly higher than the value referred to above,and the information presented should not be applied to such inputs. The effect of discontinuous acceleration inputs is considered later.
  29. 29. Generation of results • General form of spectrum is as shown in next slide (a) It approaches V= y0 at the extreme left; value of A  &&0 at the y extreme right ; it exhibits a hump on either side of the nearly horizontal central portion; and attains maximum values of U, V and y0 , y0 and &&0 & y A, which may be materially greater than the values of (a) It is assumed that the acceleration force of the ground motion, respectively. and hence the associated velocity and displacement forces, are smooth continuous functions. (c) The high frequency limit of the response spectrum for discontinuous acceleration inputs may be significantly higher than the value referred to above, and the information presented should not be applied to such inputs.
  30. 30. General form of spectrum
  31. 31. Acceleration spectra for elastic system - El Centro Earthquake
  32. 32. SDF systems with 10% damping subjected to El centro record Base shear coefficient, C Building Code Natural period,secs
  33. 33. Spectral Regions The characteristics of the ground motion which control the deformation of SOF systems are different for different systems and excitations. The characteristics can be defined by reference to the response spectrum for the particular ground motion under consideration . Systems the natural frequency of which corresponds to the Inclined left-hand portion of the spectrum are defined as low-frequency systems; Systems with natural frequencies corresponding to the nearly horizontal control region will be referred to as a medium-frequency systems ; and Systems with natural frequencies corresponding to the inclined right handed portion will be referred to as high-frequency systems.
  34. 34. Spectral Regions… Minor differences in these characteristics may have a significant effect on the magnitude of the deformation induced. Low frequency systems are displacement sensitive in the sense that their maximum deformation is controlled by the characteristics of the displacement trace of the ground motion and are insensitive to the characteristics of an associated velocity and displacement trace: Ground motions with significantly different acceleration and velocity traces out comparable displacement traces induce comparable maximum deformations in such systems.
  35. 35. Spectral Regions… The boundaries of the various frequency regions are different for different excitations and, for an excitation of a particular form, they are a function of the duration of the motion. It follows that a system of a given natural frequency may be displacement sensitive, velocity sensitive or acceleration sensitive depending on the characteristics of the excitation to which it is subjected .
  36. 36. Logarithmic plot of Deformation Spectra It is convenient to display the spectra or a log-log paper, with the abscissa representing the natural frequency of the system,f, (or some dimensionless measure of it) and the ordinate representing the pseudo velocity ,V (in a dimensional or dimensionless form). On such a plot ,diagonal lines extending upward from left to right represent constant values of U, and diagonal lines extending downward from left to right represent constant values of A. From a single plot of this type it is thus possible to read the values of all three quantities. Advantages: • The response spectrum can be approximated more readily and accurately in terms of all three quantities rather than in terms of a single quantity and an arithmetic plot. • In certain regions of the spectrum the spectral deformations can more conveniently be expressed indirectly in terms of V or A rather than directly in terms of U. All these values can be read off directly from the logarithmic plot.
  37. 37. Logarithmic plot of Deformation Spectra Velocity sensitive Displacement sensitive V0 D0 y0 & V y0 y &&0 Acceleration sensitive Log scale A0 U A Natural Frequency, f (Log scale) General form of spectrum
  38. 38. Deformation Spectra for Half-cycle Acceleration pulse: This class of excitation is associated with a finite terminal velocity and with a displacement that increases linearly after the end of the pulse. Although it is of no interest in study of ground shock and earthquakes ,being the simplest form of acceleration diagram possible ,it is desirable to investigate its effect. When plotted on a logarithmic paper, the spectrum for the half sine acceleration pulse approaches asymptotically on the left the value. V  yo& This result follows from the following expression presented earlier for fixed base systems subjected to an impulsive force, I X max  mp t1 where I   P (t ) dt 0
  39. 39. t1 Letting P(t )   m &&(t ) and y X max  U and noting that  &&(t ) dt  y 0 y & o y & we obtain, U  o or V  y o & p ( This result can also be determined by considering the effect of an instantaneous velocity change, yo ,i.e. an acceleration pulse of finite & magnitude but zero duration. The response of the system in this case is given by, uo & u(t )  uo cos pt  sin pt p Considering that the system is initially at rest, we conclude that, uo  0 and uo   yo & & yo & where, u(t )   sin pt p The maximum value of u(t), without regards to signs, is yo & U  or V  y o ) & p
  40. 40. Spectra for maximum and minimum accelerations of the mass (undamped elastic systems subjected to a Half cycle Acceleration pulse)
  41. 41. Spectra for maximum and minimum acceleration of the mass (undamped Elastic systems subjected to a versed-sine velocity pulse)
  42. 42. Deformation spectra for undamped elastic systems subjected to a versed-sine velocity pulse
  43. 43. ‘B’ Level Earthquake (=10% ; μ=1.0)
  44. 44. Deformation spectrum for undamped Elastic systems subjected to a half-sine acceleration pulse
  45. 45. Example: For a SDF undamped system with a natural frequency,f=2cps,evaluate the maximum value of the deformation,U when subjected to the half sine acceleration pulse. Assume that &&0  0.5 g ,t1=0.1sec. Evaluate y also the equivalent lateral force coefficient C, and the maximum spring force,Q0 ft1= 2 x 0.1 = 0.2 From the spectrum, . V ; y0
  46. 46. Therefore 2 .. 2 1 2p fU ᄏ f1 y 0  ᄡ 0.1ᄡ ᄡ 9.81 p p 2 1 0.1 1 U ᄏ 2 ᄡ ᄡ ᄡ 9.81  0.024 p 2 2 2 .. 4p ᄡ ᄡ t 1ᄡ y0 . A 2p fV 2p ᄡ 2 ᄡ y0 p 8t1 ᄡ 0.5 g C      8 ᄡ 0.1ᄡ 0.5  0.4 g g g g g Q0  CW  0.4W A Alternatively,one can start reading the value .. from the spectrum y0 proceeding this may, we find that A ..  0.5 y0
  47. 47. A 0.8  1 2  g Accordingly, C    0.4 g g Q0  0.4W .. A 0.8 y0 0.8  0.5  9.81 and U     0.024 m p 2 p 2 4p  2 2 2 V A The value of . and .. as read from the spectrum are y0 y0 A approximate. The exact value of .. determined is y0 0.7. This leads to C  0.385 Q0  0.385W and U  0.025
  48. 48. If the duration of the pulse were t1 = 0.75 sec instead of 0.1 sec , the results would be as follows ft1  2 ᄡ 0.75  1.5 therefore, A  1.5 .. y0 A 1.5 ᄡ 0.5 g C   0.75 g g Q0  0.75W A 1.5 ᄡ 0.5 ᄡ 9.81 U   0.047 m p 2 4p ᄡ 2 2 2
  49. 49. If the duration of the pulse were t1,as in the first case, but the natural frequency of the system were 15 cps instead of 2 cps, the results would have been as follows: ft1=15 * 0.1=1.5 A A  1.5 C  0.5  1.5  0.75 Therefore, y && g Q  0.75W A 0.75  9.81 and U 2   0.00082m 4p   15 2 p 2
  50. 50. • Plot spectra for inputs considered in the illustrative example and compare y0 For t1=0.75sec & y0 For t1=0.1sec & V .. y0 Same as in both cases f • The spectrum for the longer pulse will be shifted upward and to the left by a factor of 0.75/0.10 = 7.5
  51. 51. Design Spectrum xmax A May be determined from the spectrum by interpreting as &&  xst  0 y When displayed on a logarithmic paper with the ordinate representing V and the abscissa f, this spectrum may be approximated as follows: (Log scale) =1.5 (Log scale)
  52. 52. Deformation Spectra for Half-Cycle Velocity Pulses Refer to spectrum for   0 Note the following • At extreme right A  &&0 . Explain why? y • Frequency value behind which A  &&0 is given by ftr= 1.5 y • y The peak value of A=2 x 1.6 &&0 Explain why? In general for pulses of the same shape and duration with different n peak values A   ( &&0 ) 2j j 1 y • If duration on materially different
  53. 53. be conservative. Improved estimate may be obtained by considering relative durations of the individual pulses and superposing the peak component effects.The peak value of V is about 1.6 yo It can be shown that the absolute maximum value of the amplification factor V y0 for a system subjected to a velocity trace of a given shape is 0 approximately the same as the absolute maximum value of A0 &&0 for an y acceleration input of the same shape. This relationship is exact when the maximum response is attained following application of the pulse. But it is valid approximately even when the peak responses occur in the forced vibration era. The maximum value of U is yo and the spectrum is bounded on the left by the diagonal line U = yo
  54. 54. It should be clear that, (c) The left-hand, inclined portion of the spectrum to displacement sensitive. (e) The middle, nearly horizontal region of the spectrum is governed by the peak value of the velocity trace. It is insensitive to the shape of the pulse which can more clearly be seen in the acceleration trace. (c) The right hand portion is clearly depended on the detailed features of the acceleration trace of the ground motion. In all cases, the limiting value of on the right is equal to t1 / td.These limits appear different in the figure because of the way in which the results have been normalized. Note that the abscissa is non-dimensionalised and the ordinate with respect to the total duration of the pulse and the ordinate with respect to the maximum ground velocity. It follows that to smaller y values of &&0 corresponds to larger values of peak acceleration
  55. 55. Design Rules Design spectrum for the absolute maximum deformation of systems subjected to a half cycle velocity pulse (--undamped elastic systems;continuous input acceleration functions)
  56. 56. Deformation spectra for undamped elastic systems subjected to skewed versed-sine velocity pulses
  57. 57. Deformation Spectra for Half-cycle Displacement Pulse See spectrum for undamped systems, =0, on the next page Note that: (a) The RHS of the spectrum is as would be expected from the remarks already made. (h) Peak value of V is approximately 3.2 yo. This would be expected, as the velocity trace of the ground motion, has two identical pulses. (c) At the extreme left and of the spectrum, U=y0. The system in this case is extremely flexible and the ground displacements is literally absorbed by the spring.
  58. 58. Design Rules Design spectrum for maximum deformation of systems subjected to a half cycle displacement pulse
  59. 59. However the spectrum is no longer bounded on the left by the line U= yo, but exhibits a hump with peak value of U0 = 1.6 y0 It can be shown that the peak value of U / y0 for a system subjected to a displacement trace is approximately the same as the peak value of V / y0, induced by a velocity input of the same shape. Further more the peak value of U occurs at the same value of the dimensionless frequency parameter, f1 as the peak value of V. However it is necessary to interpret t1 as the duration of the displacement pulse, rather than of that of velocity pulse.
  60. 60. Deformation spectra for damped elastic systems subjected to a half cycle displacement pulse
  61. 61. Deformation spectra for full cycle Displacement pulse
  62. 62. As would be expected ,the maximum value of U in this case is approximately 3.2 yo .Furthermore, the left hand portion of the spectrum consists of three rather than two distinct parts: (a) The part on the extreme left for which U = yo .This corresponds to the first maximum,which occurs at approximately the instant that y attains its first maximum. (f) The smooth transition curve which defines the second maximum. This maximum occurs approximately at the instant that y(t) attains its second extremum, and is numerously greater than the peak value of the second pulse of the contribution of the first pulse.
  63. 63. Effect of Discontinuous Acceleration Pulses The high frequency limit of the deformation spectrum is sensitive to whether the acceleration force of the ground motion is a continuous or discontinuous diagram. A y The limiting value given priority applies only to continuous &&0 functions The sensitivity of the high-frequency region to the detailed characteristics of the acceleration input may be appreciated by reference to the spectra given in the following these pages. These spectra provide further confirmation to the statement made previously to the effect that low-frequency and medium-frequency systems are insensitive to the characteristics of the acceleration force of the ground motion. Explain high-frequency response to discontinuous functions.
  64. 64. Deformation spectra for damped elastic systems subjected to a full cycle displacement pulse
  65. 65. Application to Complex Ground Motions • Compound Pulses • Earthquake Records Eureka record El-Centro record Design Spectrum Minimum number of parameters required to characterize the design ground motion &&, y and y y & Max values of &&, y and y y & The predominant frequency (or deviation) of the dominant pulses in The degree of periodicity for (the number of dominant pulses in) each diagram. Dependence of these characteristics on Local soil conditions Epicentral distance and Severity of ground shaking
  66. 66. Effect of damping: • Effect is different in different frequency ranges • Effect is negligible in the extremely low frequency regime (U = y0) .. and extreme high frequency ranges (A = y0). .. u + p2u = y0(t) .. .. low frequency u = y(t) .. 0 = y0 u .. high frequency p2u = A(t) = y(t) A = y0
  67. 67. Eureka, California earthquake of Dec 21,1954 S 11o E component.
  68. 68. Eureka Quake
  69. 69. Elcentro ,California Earthquake of May 18,1940,N-S component
  70. 70. V = pseudo velocity Yc Maximum Ground Velocity Undamped Natural Frequency, f, cps
  71. 71. Further discussion of Design Response Spectra The specification of the design spectrum by the procedure that has been described involves the following basic steps: 1. Estimating the maximum values of the ground acceleration, ground velocity and ground displacement. The relationship .. . between y0, y0, y0 is normally based on a statistical study of existing earthquake records. In the Newmark – Blume – Kapur paper (“Seismic Design spectra for Nuclear Power Plants”, Jr. of Power Division, ASCE, Nov 1973, pp 287-303) the following relationship is used. 0.3 m : 0.7 m/sec : 1g for rock 0.9 m : 1.2 m/sec : 1g for Alluvium
  72. 72. 1. Estimating the maximum spectral amplification factors, αD, αV, αA ; for the various parts of the spectrum. Again these may be based on statistical studies of the respective spectra corresponding to existing earthquake records. The results will be a function not only of the damping forces of the system but also of the cumulative probability level considered.
  73. 73. Following are the values proposed in a recent unpublished paper by Newmark & Hall for horizontal motions: Damping One sigma (84.1%) Median (50%) %critical αD αV αA αD αV αA 0.5 3.04 3.84 5.10 2.01 2.59 3.65 1 2.73 3.38 4.38 1.82 2.31 3.21 2 2.42 2.92 3.66 1.63 2.03 2.74 3 2.24 2.64 3.24 1.52 1.86 2.46 5 2.01 2.30 2.71 1.39 1.65 2.12 7 1.85 2.08 2.36 1.29 1.51 1.89 10 1.69 1.84 1.99 1.20 1.37 1.64 20 1.38 1.37 1.26 1.01 1.08 1.17
  74. 74. Ground Acceleration • Number of empirical relations available in literature to correlate shaking intensity with Peak Ground Acceleration (PGA) • Table on next slide gives some such values. • Notice that the table gives – Average values of PGA; real values may be higher or lower – There is considerable variation even in the average values by different empirical relations.
  75. 75. Table Average horizontal peak ground acceleration as a function of earthquake intensity Intensity (MM Acceleration (as a fraction of g) Scale) Empirical Relations Gutenberg Newmann, Trifunac and Trifunac and Newmann, Murphy and 1954 Brady, 1975 Brady, 1977 1977 (revised and Richter, (revised by by Murphy O’Brien, 1956 Murphy and and O’Brien, 1977 O’Brien, 1977) 1977) V 0.015 0.032 0.031 0.021 0.022 0.032 VI 0.032 0.064 0.061 0.046 0.053 0.056 VII 0.068 0.13 0.12 0.10 0.13 0.10 VIII 0.146 0.26 0.24 0.23 0.30 0.18 IX 0.314 0.54 0.48 0.52 0.72 0.32
  76. 76. Ground Acceleration • ZPA stands for Zero Period Acceleration. – Implies max acceleration experienced by a structure having zero natural period (T=0). Zero Period Acceleration • An infinitely rigid structure – Has zero natural period (T=0) – Does not deform: • No relative motion between its mass and its base • Mass has same acceleration as of the ground • Hence, ZPA is same as Peak Ground Acceleration
  77. 77. Example: Determine the response spectrum for a design earthquake with &&  0.3g ye  0.3 m / sec and y0  0.25 m. Take   0.05 and use the y & amplification factors given in the preceding page. Take the knee of amplified constant acceleration point of this spectrum at 8 cps and the point beyond which A  &&0 at 25 cps y d 0.3 x 2.30 = 0.69 e 25 0.3g x 2.71 =0.813 g 50 2.3 0. = f A = 0.3g 01 2. &  y0 =0.3 m/sec 2.71 x V 2.01 25 C = 0.3 0. &&0  y =0.3g 0.3g y0=0.25 m Q = 0.3W Y=0.00127   0.05 0.22 cps 1.81 cps 8 cps 25 cps f Note: In the spectra recommended in the Newmark – Blume -Kapur paper, the line de slope upward to the left and the line of slopes further downward to the right
  78. 78. Design Earthquakes Describing the Earthquake Ground Motion Time Histories  Ground motion time histories are numerical descriptions of how a certain ground motion parameter, such as acceleration, varies with time.  They provide a full description of the earthquake motion, unlike response spectra, as they show duration as well as amplitude and frequency content.  They are usually expressed as plots of the ground motion parameter versus time, but consist of discrete parameter-time pairs of values.  Idealized time histories are sometimes represented by simple mathematical functions such as sine waves, but real earthquake motions are far too complex to be represented mathematically.  There are two general types of time histories: - Recorded (often referred to as historical records) - Artificial
  79. 79. Statistically Derived Design Spectra  The general procedure for generating statistically derived spectra is as follows:  Classes of ground motions are selected (based on soil, magnitude, distance, etc.)  Response spectra for a large number of corresponding ground motions are generated and averaged  Curves are fit to match computed mean spectra  Resulting equations are used to develop a design response spectrum with desired probability of exceedence
  80. 80. Effect of various factors on spectral values Soil Conditions  For soft soils, ag remains the same or decreases relative to firm soil,but vg and dg increase, generally.  Layers of soft clay, such as the Young Bay Mud found in the San Francisco Bay area, can also act as a filter, and will amplify motion at the period close to the natural period of the soil deposit.  Layers of deep, stiff clay can also have a large effect on site response.  For more information on site effects, see Geotechnical Earthquake Engineering by Kramer.
  81. 81. Effect of various factors on spectral values Near Fault Motions and Fault Rupture Directivity For near-fault motions ag increases, but vg increases more dramatically due to effect of a long period pulse. This pulse is generally most severe in the fault normal direction (as it can cause fling), but significant displacement also occurs in the fault parallel direction. The fault parallel direction usually has much lower spectral acceleration and velocity values than the fault normal direction. Sample waveforms are located in a previous section of the notes, Factors Influencing Motion at a Site. No matter the directivity, however, the motions very close to the fault rupture tend to be more severe than those located at moderate distances.
  82. 82. Effect of various factors on spectral values Near Fault Motions and Fault Rupture Directivity (Cont..) Somerville et al. have developed a relationship which converts mean spectral values generated from attenuation relationships to either the fault parallel or fault normal component of ground motion. See the shift of the spectrum in the long period range.
  83. 83. Effect of various factors on spectral values Viscous Damping Friction between and with structural and non-structural elements Localized yielding due to stress concentrations and residual stresses under low loading and gross yielding under higher loads Energy radiation through foundation Aeroelastic damping Viscous damping Analytical modeling errors
  84. 84. Effect of various factors on spectral values Viscous Damping  Viscous Damping Values for Design  Many codes stipulate 5% viscous damping unless a more properly substantiated value can be used.  Note that actual damping values for many systems, even at higher levels of excitation are less than 5%.
  85. 85. Effect of Various Factors on Spectral Values Modifying the Viscous Damping of Spectra Newmark and Hall's Method For each range of the spectrum, the spectral values are multiplied by the ratio of the response amplification factor for the desired level of damping to the response amplification factor for the current level of damping.  Consider if we have a median spectrum at 5% viscous damping and we would like it at x%.  If the 5% Joyner and Boore Sv value is 60 cm/sec on the descending branch, an estimate of the 2% Sv value is 60x(2.03/1.65) = change 60x1.47 = 88 cm/sec
  86. 86. Role of Ductility
  87. 87. Elasto Plastic Force Elasto Plastic system and its Deformation relation corresponding linear system
  88. 88. Design Values of normalized yield Strength
  89. 89. Construction of Elastic Design Spectrum
  90. 90. Construction of Inelastic Design Spectrum
  91. 91. Response of Elastoplastic system to Elcentro Ground motion
  92. 92. Empirically Derived Design Spectra Basic Concepts  The complexity of the previous methods, and the limited number of records available two decades ago, led many investigators to develop empirical methods for developing design spectrum from estimates of peak or effective ground motion parameters.  These relationships are based on the concept that all spectra have a characteristic shape, which is shown here.
  93. 93. Empirically Derived Design Spectra Newmark and Hall's Method  N. M. Newmark and W. J. Hall's procedure for developing elastic design spectra starts with the peak values of ground acceleration, velocity, and displacement.  These values are used to generate a baseline curve that the spectrum will be generated from.  The values of peak ground acceleration and velocity should be obtained from a A typical baseline curve plotted on deterministic or probabilistic seismic hazard tripartite axes is shown above. analysis  The value of peak ground displacement is a bit more difficult to obtain due to the lack of reliable attenuation relationships.  Some empirical functions utilizing the PGA are available to provide additional estimates of the peak ground displacement.
  94. 94. Empirically Derived Design Spectra (Cont..) Newmark and Hall's Method Structural Response Amplification Factors Structural response amplification factors are then applied to the different period-dependent regions of the baseline curve Structural response amplification factors Damping Median + One Sigma (% critical) a v d a v d 1 3.21 2.31 1.82 4.38 3.38 2.73 2 2.74 2.03 1.63 3.66 2.92 2.42 3 2.46 1.86 1.52 3.24 2.64 2.24 5 2.12 1.65 1.39 2.71 2.3 2.01 7 1.89 1.51 1.29 2.36 2.08 1.85 10 1.64 1.37 1.2 1.99 1.84 1.69 20 1.17 1.08 1.01 1.26 1.37 1.38
  95. 95. Empirically Derived Design Spectra (Cont..) Newmark and Hall's Method Tripartite Plots: Newmark and Hall's spectra are plotted on a four-way log plot called a tripartite plot. This is made possible by the simple relation between spectral acceleration, velocity, and displacement: Sa/w = Sv = Sdw A tripartite plot begins as a log-log plot of spectral velocity versus period as shown.
  96. 96. Empirically Derived Design Spectra (Cont..) Newmark and Hall's Method  Then spectral acceleration and spectral displacement axes are superimposed on the plot at 45 degree angles
  97. 97. Empirically Derived Design Spectra (Cont..) Newmark and Hall's Method  All three types of spectrum (Sa vs. T, Sv vs. T, and Sd vs. T) can be plotted as a single graph, and three spectral values for a particular period can easily be determined.  The Sa, Sv, and Sd values for a period of 1 second are shown below.
  98. 98. Empirically Derived Design Spectra Constructing Newmark and Hall Spectra 1. Construct ground motion 'backbone' curve using constant agmax, vgmax, dgmax lines. Take lower bound on three curves (solid line on figure)                                                              

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